Benadryl once required a prescription. So did Claritin. They all carry lists of warnings.
The FDA made these OTC drugs only after many years and millions of doses leading to the conclusion that they were safe enough to be taken without a physicians supervision. The fact that Diamox has not says that the FDA has not determined that yet (though it may be simply because the prescribing volume of Diamox is too low to get enough data).
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Actually, and it is an interesting point, that is not the pathway by which a prescription drug goes over the counter. What is required is for the manufacturer to submit a request, along with supporting studies to the FDA, which then considers the conversion. This process costs many millions of dollars. Unless a manufacterer can reasonably expect to recoup that cost, they won't do it, so it won't happen.
I have no doubt that Diamox would get such approval, particularly in the very low doses used for AMS. In comparison, there are things out there approved that would NEVER get approval. It may interest folks to know that the most common cause of poisoning deaths of children in the US is by vitamins.
http://www.cfsan.fda.gov/~dms/bgiron.html